I did this over the weekend. I used a 2 and a half foot concrete stake and wedged the pointed end between the transmission and the axle casing. I tapped the other end of the concrete stake with a hammer about medium strength. Because the axle casing is beveled the ring popped right out with only about 3 hits. The axle casing slid right out after that. I was really expecting a huge battle, but was amazed how easily it came out.
Putting the new axle on was not much more difficult. I fed the spline in the transmission and then pulled it back about 4 inches and rammed it in. This also went in on the 3rd try.
Altogether the project went far smoother than I was expecting and I am confident in tackling the other side.
Good luck!